Mathematics – Algebraic Geometry
Scientific paper
2008-12-19
Mathematics
Algebraic Geometry
42 pages, 17 Postscript figures
Scientific paper
This article tries to answer the question: How far can the algebro-geometric theory of rational descendant Gromov-Witten invariants be carried over to the tropical world? Given the fact that our moduli spaces are non-compact, the answer is surprisingly positive: We discuss universal families and the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we give two criteria that suffice to prove the tropical version of a particular WDVV or topological recursion equation. Discussing these criteria in the case of curves in R^1 or R^2, we prove, for example, that for the toric varieties P^1, P^2, P^1 \times P^1, F_1, Bl_2(P^2), Bl_3(P^2) and with Psi-conditions only in combination with point conditions, the tropical and conventional descendant Gromov-Witten invariants coincide. In particular, we can unify and simplify the proofs of the previous tropical enumerative results.
Rau Johannes
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